Title: The power of approximation theory: from splines to introducing a new class of functions  

Abstract: The power of approximation theory is in representing complex functions or data with simpler, more manageable forms. The primary goal of approximation theory is to find suitable approximations that capture essential characteristics of the original object, facilitating analysis and computation via means of polynomials, splines, wavelets, etc. The applications of approximation theory span numerous fields, such as signal processing, image compression, numerical analysis, and scientific computing, providing valuable insights into the balance between accuracy and simplicity in representing real-world phenomena. In my talk, I will discuss the power of approximation theory, introduce a new class of functions, and discuss its properties. Specifically, I will rigorously show that it is Holder continuous, and has Holder continuity of the highest-order well-defined derivatives.